More Properties of Functions …in Sec. 1.2b Homework: p. 98 25-61 odd

Let’s see these graphically… Definition: Increasing, Decreasing, and Constant Function A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x). Let’s see these graphically…

Increasing Decreasing Increasing on Constant on a b Constant Decreasing on

Definition: Lower Bound, Upper Bound, and Bounded A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above if there is some number B that is greater than or equal to every number in the range of f. Any such number B is called an upper bound of f. A function f is bounded if it is bounded both above and below. Let’s see these graphically…

Only bounded below Not bounded above or below Only bounded above Bounded

Definition: Local and Absolute Extrema A local maximum of a function f is a value of f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f. A local minimum of a function f is a value of f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f. Local extrema are also called relative extrema. Let’s see these graphically…

Maximum Local Maximum Local Minimum Any Absolute Minima?

Guided Practice Decreasing on Increasing on For the given function, identify the intervals on which it is increasing and the intervals on which it is decreasing. Decreasing on Increasing on

Whiteboard practice: Increasing on and Decreasing on and For the given function, identify the intervals on which it is increasing and the intervals on which it is decreasing. Increasing on and Decreasing on and

Whiteboard practice: Bounded Identify the given function as bounded below, bounded above, or bounded. Bounded

Min of –24.057 at x = –2.057, Local Min of –1.766 Whiteboard practice: Decide whether the given function has any extrema. If so, identify each maximum and/or minimum. Min of –24.057 at x = –2.057, Local Min of –1.766 at x = 1.601, Local Max of 1.324 at x = 0.456

We will look at three types of symmetry: 1. Symmetry with respect to the y-axis Ex: f(x) = x 2 For all x in the domain of f, f(–x) = f(x) Functions with this property are called even functions

We will look at three types of symmetry: 2. Symmetry with respect to the x-axis Ex: x = y 2 Graphs with this symmetry are not functions, but we can say that (x, –y) is on the graph whenever (x, y) is on the graph

We will look at three types of symmetry: 3. Symmetry with respect to the origin Ex: f(x) = x 3 For all x in the domain of f, f(–x) = – f(x) Functions with this property are called odd functions

Guided Practice  Even  Neither Tell whether each of the following functions is odd, even, or neither (solve graphically and algebraically): First, check the graph, then verify algebraically…  Even  Neither

Definition: Horizontal and Vertical Asymptotes The line y = b is a horizontal asymptote of the graph of a function y = f(x) if f(x) approaches a limit of b as x approaches + or – . In limit notation: 8 8 or

Definition: Horizontal and Vertical Asymptotes The line x = a is a vertical asymptote of the graph of a function y = f(x) if f(x) approaches a limit of + or – as x approaches a from either direction. In limit notation: 8 8 or

Vertical Asymptotes at x = –1 and x = 2, Horizontal Asymptote at y = 0 Guided Practice Identify any horizontal or vertical asymptotes of the graph of: Vertical Asymptotes at x = –1 and x = 2, Horizontal Asymptote at y = 0

End Behavior Sometimes, it is useful to consider the “end behavior” of a function. That is, what does the function look like as the dependent variable (x) approaches infinity?

Whiteboard practice: Using any method, find all asymptotes.